Improved universal approximation with neural networks studied via affine-invariant subspaces of $L_2(\mathbb{R}^n)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909563384496128 |
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| author | Schneider, Cornelia Probst, Samuel |
| author_facet | Schneider, Cornelia Probst, Samuel |
| contents | We show that there are no non-trivial closed subspaces of $L_2(\mathbb{R}^n)$ that are invariant under invertible affine transformations. We apply this result to neural networks showing that any nonzero $L_2(\mathbb{R})$ function is an adequate activation function in a one hidden layer neural network in order to approximate every function in $L_2(\mathbb{R})$ with any desired accuracy. This generalizes the universal approximation properties of neural networks in $L_2(\mathbb{R})$ related to Wiener's Tauberian Theorems. Our results extend to the spaces $L_p(\mathbb{R})$ with $p>1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02445 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved universal approximation with neural networks studied via affine-invariant subspaces of $L_2(\mathbb{R}^n)$ Schneider, Cornelia Probst, Samuel Functional Analysis Information Theory We show that there are no non-trivial closed subspaces of $L_2(\mathbb{R}^n)$ that are invariant under invertible affine transformations. We apply this result to neural networks showing that any nonzero $L_2(\mathbb{R})$ function is an adequate activation function in a one hidden layer neural network in order to approximate every function in $L_2(\mathbb{R})$ with any desired accuracy. This generalizes the universal approximation properties of neural networks in $L_2(\mathbb{R})$ related to Wiener's Tauberian Theorems. Our results extend to the spaces $L_p(\mathbb{R})$ with $p>1$. |
| title | Improved universal approximation with neural networks studied via affine-invariant subspaces of $L_2(\mathbb{R}^n)$ |
| topic | Functional Analysis Information Theory |
| url | https://arxiv.org/abs/2504.02445 |